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Master Thesis Presentation, 14Dec07
Pair Wise Distance Histogram Based Fingerprint Minutiae Matching Algorithm
Developed By: Neeraj Sharma M.S. student, Dongseo University, Pusan South Korea.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Contents Introduction Why Fingerprints: some facts Essential Preprocessing (Feature Extraction etc.) Abstract Previous Work Problem Simulation Steps of Algorithm Flow Chart Local Matching Global Matching Results Comparison with Reference method (Wamelen et al) Future work Publications
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Introduction Fingerprints are most useful biometric feature
in our body. Due to their durability, stability and uniqueness
fingerprints are considered the best passwords. In places of access security, high degree
authentication, and restricted entry, fingerprints suggests easy and cheap solutions.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Market Capture by different Biometric modalities
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Different fingerprints of two fingers
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Different Features in a Fingerprint
Ridge Ending
Enclosure
Bifurcation
Island
Texture
Singular points
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Extraction of minutiae
Image skeleton
Gray scale image
Minutia features
Feature Extraction with CUBS-2005 algorithm (Developed by SHARAT et al)
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Feature points pattern of same finger
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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High level description of algorithms in FVC (Fingerprint Verification Competition)2004
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Abstract
Thesis proposes a novel approach for matching of minutiae points in fingerprint patterns.
The key concept used in the approach is the neighborhood properties for each of the minutiae points.
One of those characteristics is pair wise distance histogram, that remains consistent after the addition of noise and changes too.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Previous Work Fingerprint Identification is quite mature area
of research. Its almost impossible to describe all the previous approaches in a short time here.
The previous methods closely related to this approach and also taken in reference are by Park et al.[2005] and Wamelen et al.[2004].
Park et al. used pair wise distances first ever to match fingerprints in their approach before two years.
Wamelen et al. gave the concept of matching in two steps, Local match and Global match.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Problem Simulation The input fingerprint of the
same finger seems to be different while taken on different times.
There may be some translational, rotational or scaling changes, depending upon situation.
Our aim is to calculate these changes as a composite transformation parameter “T”.
The verification is done after transforming the input with these parameters, new transformed pattern should satisfy desired degree closeness with template pattern.
Template Input Pattern
cos sin( )
sin cosq x p
q y p
x t xs sq T p
y t ys s
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Minutiae matching-Aligning two point sets
Input
Template
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Algorithm Steps The algorithm runs in two main steps:- (i) Local matching (ii) Global matchingIn local matching stepwise calculations are there:1. Calculate “k” nearest neighbors for each and
every point in both patterns.2. Calculate histogram of pair wise distances in
the neighborhood of every point.3. Find out the average histogram difference
between all the possible cases.4. Set the threshold level of average histogram
difference5. Compare the average histogram differences
with the threshold level.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Flow Chart
No
No
Point pattern “P” stored in database pattern “Q” is taken that is to be matched with “P”
Select a local point and it’s “k” nearest neighbors in patterns P
Select a local point and it’s “k” nearest neighbors in patterns Q
Make pair wise distance Histogram
Average histogram difference < Threshold level
Calculate and store transformation parameter
Iteration algorithm to calculate final Transformation Parameter
Start
All point’s in pattern p is examined
End
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Calculation of “k” Nearest neighbors (Local match)
For the given input fingerprint pattern and the template pattern, calculate “k” nearest neighbors in order to distances.
Here k is a constant can be calculated with the formula given by wamelen et al.(2004)
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Histogram Calculation (Local Match) Histogram of pair wise distances in
their neighborhood for each and every point is calculated here. It describe the variety of distances of particular point in its neighborhood.
Here for one point “P1”;
P1n1, P1n2, P1n3, P1n4, P1n5 are five nearest neighbors.
Note: step size is 0.04unit, here.
P1n2
P1
P1n1
P1n2
P1n3
P1n4
P1n5
P1n1
P1n3
P1n4
P1n5
P1n2
P1n3
P1n4
P1n5
P1n3
P1n4
P1n5
P1n4 P1n5
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Average Histogram Difference and Threshold Setting (Local Match)
To calculate average histogram differences for two points, first subtract the their histograms. It comes in a form of matrix. To calculate average, just normalize it on corresponding scale.
H1=[4 2 0 0 2 0 1 2 3 1] H2=[1 3 2 5 0 0 2 3 0 1] H1 - H2 =[3 -1 -2 -5 2 0 -1 -1 3 0] Average histogram diff.(ΔH avg)=(1/10)*Σ(| H1 - H2 |i) Setting of threshold depends on the size of point pattern.
Larger the number of points, smaller the threshold count. Every matching pair is related with a transformation
function. That transformation parameter is calculated mathematically.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Threshold check for a 20 points pattern input0 1.2 2 2.4 1.2 1.6 0.8 1.6 1.6 0.8 2.8 2 2.8 2.4 2.8 2.4 1.6 2.8 2.8 2.8 1
1.2 0.4 1.2 1.6 0.8 1.2 1.2 1.6 2 1.2 2.8 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 2 2
2 1.2 0.8 1.2 1.6 1.2 2.4 2.4 2.4 1.6 2.4 1.2 1.2 1.2 1.6 1.6 2.8 1.6 0.8 1.6 3
2.4 1.6 0.8 0.4 1.6 0.8 2.4 2 2 1.6 2 0.8 1.2 1.6 1.2 1.6 2 1.2 1.6 1.6 4
1.2 0.8 1.6 2 0.4 1.6 0.8 1.2 1.6 1.2 2.4 1.6 1.6 1.6 1.6 1.6 1.2 1.6 2 2 5
1.6 0.8 0.8 0.8 1.2 0 2.4 1.6 1.6 0.8 2 1.2 1.6 1.6 1.6 1.6 1.6 1.6 2 1.6 6
0.8 1.6 2.4 2.8 1.2 2.4 0 1.6 2 1.6 3.2 2.4 2.4 2.4 2.4 2.4 1.6 2.4 2.8 2.8 7
1.6 1.2 2 2.4 0.8 1.6 1.6 0 0.8 1.6 2 2.4 2.4 2 1.6 1.2 1.6 1.6 2.8 1.6 8
1.6 1.6 2 2.4 1.2 1.6 2 0.8 0 1.6 2 2.4 2.8 2 2.4 2 2 2.4 2.8 2.4 9
0.8 0.8 1.6 1.6 0.8 0.8 1.6 1.6 1.6 0 2 1.2 2 2 2 2 1.6 2 2.4 2 10
2.8 2.4 2.4 2.4 2 2 3.2 2 2 2 0 2.4 2.4 1.6 1.6 1.2 2.4 2 2.8 1.6 11
2 1.6 0.8 1.2 1.2 1.2 2 2 2 1.2 2 0.4 0.8 1.6 1.6 2 1.6 1.2 2 2 12
2.8 1.6 1.2 1.6 1.6 1.6 2.4 2.4 2.8 2 2.4 1.2 0 1.2 1.6 1.6 1.6 1.2 1.6 1.6 13
2 1.2 1.6 2 0.8 1.6 2 1.6 2 1.6 1.6 1.6 1.2 0.4 2 1.2 2 2 1.6 1.6 14
2.8 1.6 1.6 1.6 1.6 1.6 2.4 1.6 2.4 2 1.6 2 1.6 2 0 0.8 2 0.4 2 0.8 15
2.8 1.6 1.2 2 1.6 2 2.4 1.6 2.4 2.4 1.6 2 1.2 1.6 0.8 0.4 2 0.8 2 1.2 16
1.6 1.6 2 2.4 1.2 1.6 1.6 1.6 2 1.6 2.4 2 1.6 2.4 2 2 0 1.6 3.2 2 17
2.4 2 2.4 2.8 1.6 2.4 2.4 1.6 2.4 2 1.6 2 1.2 1.6 2 1.6 1.6 1.6 2.8 1.6 18
2.8 1.6 1.6 1.6 1.6 1.6 2.8 2 2 2.4 2 2 1.6 0.8 1.6 1.2 2.4 1.6 0.8 1.2 19
2.8 1.6 2 2 1.6 1.6 2.8 1.6 2.4 2 1.6 2.4 1.6 1.6 0.8 0.8 2 0.8 2 0 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Transformation Parameter calculation On the basis of histogram differences, we can make decision on
local matching pairs. Then the transformation parameter is calculated in the following way by least squire method. Here “r” represents the corresponding Transformation Parameter.
0
010
0
A xA yA xB
A yA xA yB
xA yA A B
yA xA A B
l
lr
l lD
l l
1i
k
xA ai
x
1
i
k
xB bi
x
1i
k
yA ai
y
1
i
k
yB bi
y
1
i i i i
k
A B a b a bi
l x x y y
1
i i i i
k
A B a b a bi
l x y y x
2 2
1i i
k
A a ai
l x y
2 2A xA yAD ll
cos sinT
x yr t t s s
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Axial Representation of all Transformation Parameters after Local match
Three axes represent the translational (in both x& y direction), rotaional and scale changes.
The most dense part in graph represents the correct transformation parameters only. We need to conclude our results to that part.
-100-50
050
100
1.5
2
2.5
32
2.5
3
3.5
4
trans
all iterations together 5
rotangle
scale
-100
-80
-60
-40
-20
0
20
40
60
80
100
1.5
2
2.5
3
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
trans
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Mean and Standard Deviation
Mean and standard deviation is calculated with the following mathematical equations.
n
iixnx
1
/1
N
ii xx
N 1
2)(1
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Global Matching (Iteration Algorithm) Iteration method is used to
converge the result towards dense part of the graph.
For applying this method we need to calculate mean and standard deviation of the distribution.
In the graph all transformation parameters are present, calculated after local matching step.
The mean for this distribution is shown by the “triangle” in centre.
-200-100
0100
200
010
2030
40500
0.5
1
1.5
2
2.5
trans
All transformation-parameters After Local Matching and Threshold Test
rotangle
scal
e
High Density Area
Mean
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Result after first iteration
In this graph, black triangle is describing the mean for the distribution.
After one iteration step some of the transformation parameters, due to false local match got removed.
-200-100
0100
200
010
2030
40500
0.5
1
1.5
2
2.5
trans
Transformation Parameters After First Iteration Step
rotangle
scal
e
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Result after second iteration
After second iteration, mean converges more towards the dense area.
Black triangle is the mean point for this distribution shown here.
-200-100
0100
200
010
2030
40500
0.5
1
1.5
2
2.5
trans
Transformation Parameters After Second Iteration Step
rotangle
scal
e
Mean converges to density
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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After third iteration
Performing iterations to converge the result, gives the distribution having least standard deviation.
Black star in this graph is the desired transformation parameter i.e. “r”
cos sinT
x yr t t s s -200
-1000
100200
010
2030
40500
0.5
1
1.5
2
2.5
trans
Transformation parameters after many iterations having minimum Std.Deviation
rotangle
scal
e
Parameters and their mean final stage
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Verification by transforming the template with calculated parameters
Template Pattern in Database Transformed version with Parameter ”r”
Verification by Overlapping with original input pattern
Transformed With parameter “r”
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Matching Result for Ideal point sets No missing point N=60 Exactly matching
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
neighbours of point P
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4neighbours of point q
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Random and Normalized Noise pattern
(a)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
1
2
3
4
5
6
7
8
9
Magnitude of added noise in x & y direction
frequen
cy o
f part
icul
ar
magnitude t
o o
ccur
Histogram pattern of Random noise
noise in x-dir
noise in y-dir
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050
2
4
6
8
10
12
magnitude of noise added
frequen
cy o
f th
e p
art
icula
r nois
e m
agnitude
Histogram pattern of normalized external noise
noise in x-dir
noise in y-dir
0 5 10 15 20 25 30 35 40 45 50-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Index of point
valu
e o
f nois
e a
dded in x
& y
direction f
or
each p
oin
t
Plot for values of noise in each point
noise in x
noise in y
0 5 10 15 20 25 30 35 40 45 500
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Valu
es o
f nois
e a
dded in x
& y
direction f
or
each p
oin
t
Index of particular point
Plot of random noise added in both direction
(b)
(c) (d)
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Results With Randomly Missing Points
Missing points =20 Total points=60 Matching
points=36 Matching factor=
0.76 Noise factor=0.031
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4neighbours of point P
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
neighbours of point q
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Matching results after missing points
Half no. of points (30) missing from pattern
Missing points =30 Total points=60 Matching
points=16 Matching factor=
0.91 Noise factor=0.021
0 0.2 0.4 0.6 0.8 1 1.2 1.4
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
neighbours of point P
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5neighbours of point q
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Definitions t= λ *r/(2*√n) t is distance of closeness which can be some fraction
of minimum pair-wise distance λ is called “matching factor”, depends on point
pattern r=maximum pair-wise distance/2 N is no. of points in the pattern η is “noise factor” shows the extent of noise added to
point pattern η= added average error/mean pair-wise distance
Master Thesis Presentation, 14Dec07
Results for different λ and ηTotal points Matching
factor(λ)Noise factor(η)
Time of match(sec)
Accuracy(%)
50 0.508 0.024 1.97 99
60 1.12 0.016 2.53 98
70 4.73 0.019 3.12 94
70 1.23 0.0367 3.10 92
80 3.67 0.0198 4.01 93
80 2.78 0.0276 3.80 90
80 0.8954 0.0431 4.00 89
90 6.01 0.019 5.00 98
90 2.75 0.026 4.84 95
90 2.22 0.0398 4.93 90
90 0.8551 0.049 4.86 83
100 0.6159 0.018 6.063 97
100 2.2081 0.0226 6.00 93
100 3.2826 0.035 5.86 90
100 1.43 0.039 6.00 88
100 2.40 0.0435 6.016 85
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Performance with missing points regionally
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2original pointset "p"
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2pointset "q" after missing region and scaling, rotation changes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1overlapping of P and Q pattern at 0.4 diff 3d ref
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2removed elliptical region from original pointset "p"
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Performance with missing points regionally
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1overlapping of P and Q pattern at 0.4 diff 4d ref
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3original pointset "p"
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3neighbours of point q
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3removed elliptical region from original pointset "p"
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Performance with missing points regionally
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3original pointset "p"
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3neighbours of point q
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3removed elliptical region from original pointset "p"
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9overlapping of P and Q pattern at 0.4 diff 4d ref
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Performance with a real fingerprint
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Template Pattern "P"
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4neighbours of point qInput realfingerprint "Q"
0.5 1 1.5 2 2.5 32
2.5
3
3.5
4
4.5
5Verfication of matching by overapping of P and Q pattern
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Results with real and random fingerprints
The algorithm was tested on both randomly generated point pattern and real data base.
The results shows correct identification in more than 93.73% cases out of 500 tests, with randomly generated data.
For real fingerprint data, method was tested on some FVC (Fingerprint Verification Competition) 2004 samples. In most of the cases performance was found satisfactory.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Comparative Performances of two methods over randomly data
Total points to match
Missing Points External noise added (%)
Translation [x y]
Time of match with wamelen’s method (sec.)
Time of match with histogram method (sec.)
30 10 2.4 1.5 2.1 1.36 1.01
40 15 1.85 1.2 1.8 2.07 1.40
50 20 2.1 2.1 2.0 2.90 1.88
60 25 3.1 1.4 1.8 3.96 2.43
70 30 2.3 1.7 1.6 5.01 3.10
80 35 1.9 1.2 1.9 6.30 3.90
90 40 3.2 1.3 1.1 7.72 4.83
100 45 2.9 2.1 1.7 9.26 5.88
110 45 2.6 2.2 1.8 11.10 7.00
120 50 2.5 2.1 1.7 12.95 8.31
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Comparison of performances
0 50 100 150 200 2500
10
20
30
40
50
60
no of total points for match
TIm
e ta
ken
for
mat
chin
g in
sec
Comparision of performance
Wamelen's Algo
New Method
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Advantages of the Method over others Proposed earlier This algorithm undergoes two steps, so
accuracy is good and false acceptance rate is low.
Calculation is less complex with comparison to other methods proposed yet. Here, histogram is a basis to select the local matching pairs, while in other randomize algorithms are lacking in any basic attribute to compare.
Performance is better in case with missing points from a specific region.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Limitations
This algorithm is dependent on accuracy of feature extraction method used for minutiae extraction.
Method performs well if the number of missing points in the pattern is less than 50% of total minutiae points.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Future Work To enhance the performance of algorithm on
real fingerprint data is also a big challenge. To calculate the computational complexity in
big “O” notation. One important task is to develop an
independent method for feature points extraction from fingerprints.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Publications
Journal:1. Sharma Neeraj, Lee Joon Jae “Fingerprint Minutiae
Matching Algorithm Using Distance Histogram Of Neighborhood”, Journal of KMMS. (To be published in Dec. 2007 edition)
International Conferences:1. Sharma Neeraj, Choi Nam Seok, Lee Joon Jae,
“Fingerprint Minutiae Matching Algorithm Using Distance Distribution Of Neighborhood”, MITA (2007), 21-24.
2. Lye Wei Shi, Sharma Neeraj, Choi Nam Seok, Lee Joon Jae, Lee Byung Gook, “Matching Of Point Patterns By Unit Circle”, APIS(2007), 263-266.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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References1. Wamelen P. B. Van, Li Z., and Iyengar S. S.: “A fast expected time algorithm
for the 2-D point pattern matching problem. Pattern Recognition” 37, Elsevier Ltd, (2004), 1699-1711.
2. Park Chul-Hyun, Smith Mark J.T., Boutin Mireille, Lee J.J.: “Fingerprint Matching Using the Distribution of the Pairwise Distance Between Minutiae”, AVBPA (2005), LNCS 3546 (2005), 693-701.
3. Sakata Koji, Maeda Takuji, Matsushita Masahito, Sasakawa Koichi, Tamaki Hishashi: “Fingerprint Authentication based on matching scores with other data”, ICB, LNCS 3832, (2006), 280-286.
4. Maltoni D., Maio D., Jain A.K., Prabhakar S. :”Handbook of Fingerprint Recognition”, Springer 2003.
5. Chang S.H., Cheng F. H., Hsu Wen-Hsing, Wu Guo-Zua: “Fast algorithm for point pattern matching: Invariant to translation, rotations and scale changes.” Pattern Recognition, Elsevier Ltd., Vol-30, No.-2, (1997), 311-320.
6. Irani S., Raghavan P.:” Combinatorial and Experimental Result on randomized point matching algorithms”, Proceeding of the 12th Annual ACM symposium on computational geometry, Philadelphia, PA, (1996), 68-77.
7. Adjeroh D.A., Nwosu K.C.: ”Multimedia Database Management – Requirements and Issues”, IEEE Multimedia. Vol. 4, No. 3, 1997, pp 24-33.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
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Thanks for your kind attention.
Master Thesis Presentation: 14DEC07Presented by:[email protected]
51
(a)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
1
2
3
4
5
6
7
8
9
Magnitude of added noise in x & y direction
frequency o
f part
icula
r m
agnitude t
o o
ccur
Histogram pattern of Random noise
noise in x-dir
noise in y-dir
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050
2
4
6
8
10
12
magnitude of noise added
frequency o
f th
e p
art
icula
r nois
e m
agnitude
Histogram pattern of normalized external noise
noise in x-dir
noise in y-dir
0 5 10 15 20 25 30 35 40 45 50-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Index of point
valu
e o
f nois
e a
dded in x
& y
direction f
or
each p
oin
t
Plot for values of noise in each point
noise in x
noise in y
0 5 10 15 20 25 30 35 40 45 500
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Valu
es o
f nois
e a
dded in x
& y
direction f
or
each p
oin
t
Index of particular point
Plot of random noise added in both direction
(b)
(c) (d)